I've created child NS for my domain, then added a DNS zone in ISPConfig. Now all sites go there.

So I got a domain with NameSilo, I created child NS (glue records) that point to my server. Let's call it mysite.com.
On my server I did what I believe is the correct DNS configuration and all is fine and dandy.
Except one thing. Now when I open any of the other sites, I see the mysite.com page. Most of the other domains rely on external DNS hosting so there are no ISPConfig DNS entries for them.